Geometric presentations for the pure braid group

TL;DR

This paper introduces new geometric finite presentations for the pure braid group using a metric on the punctured disc, emphasizing simplicity and geometric intuition, including the twist and swing presentations.

Contribution

It provides novel, easy-to-remember geometric presentations for the pure braid group, connecting them to mapping class group concepts like Dehn twists.

Findings

Presented the twist and swing geometric presentations.

Connected the swing presentation to Dehn twists and classical relations.

Simplified the understanding of pure braid group structure.

Abstract

We give several new positive finite presentations for the pure braid group that are easy to remember and simple in form. All of our presentations involve a metric on the punctured disc so that the punctures are arranged "convexly", which is why we describe them as geometric presentaitons. Motivated by a presentation for the full braid group that we call the "rotation presentation", we introduce presentations for the pure braid group that we call the "twist presentation" and the "swing presentation". From the point of view of mapping class groups, the swing presentation can be interpreted as stating that the pure braid group is generated by a finite number of Dehn twists and that the only relations needed are the disjointness relation and the lantern relation.